Top of Part 3
|3-1: Energy Eigenvalues and Eigenstates|
We learned on the page,
4-3: The Bohr Model of Atoms,
in the previous Seminar
that the atomic structure
can beautifully be described
by Bohr's old
in which it was hypothesized
(step-like) values of
energy are allowed
for the atomic stationary states
by the quantum condition.
We learned that this
old quantum theory
can well explain
of hydrogen atom.
Then, we have a question whether Bohr's old quantum theory can be derived from quantum mechanics. Does the Schroedinger equation bring out the stationary states with discrete energies?
[Energy Eigenvalues of Bound States]
We learned on the page, 1-6: Energy Eigenvalues, Eigenstates, that, if we treat a harmonic oscillator (spring vibration) with quantum mechanics, we could obtain the eigenstates with the discrete energy eigenvalues.
Such discrete or step-like eigenvalues are not only in a harmonic oscillator. These are a common property of the states called "bound states" in quantum mechanics. This property is never seen in the classical theory. What is its origin? Let us briefly explain this below.
Suppose a potential well shown in the above Fig. (A). Consider the motion of a particle of mass m satisfying the Schroedinger equation
Here we consider a state in which the particle is bound in the square-well potential. As seen in the figure, the energy E of the state must be
Let us see the feature of the wave function in the region of a long distance where the potential vanishes, i.e., V (x ) = 0. In this region, the Schroedinger equation (1) becomes
Accordingly, the general solution of the wave function in the region of V (x ) = 0 is written
where A and B are arbitrary constants.
Since the square of the absolute value of the wave function denotes the probability density that the particle will be found, the wave function must not diverge at any point in the space. Accordingly, in the wave function (4), the constant A must be 0 at the distant right and the constant B must be 0 at the distant left , i.e., the wave function must be
In order that the boundary condition that the wave functions do not diverge at is satisfied, the energy E cannot be arbitrary but must be of some specific values. Thus the allowed energies are not continuous but discrete. Such specific discrete (step-like) energies are called energy eigenvalues, which was derived from the German word eigen meaning "characteristic" or "unique". We call this type of energies discrete energy eigenvalues or say that the energy is quantized. In addition, the state corresponding to each energy eigenvalue is called the eigenstate. The wave functions of these eigenstates vanishes at the long distance (). Since almost all part of the wave functions are bound (or localized) in the potential region, these eigenstates are called bound states. We can, therefore, say that the energy eigenvalues of bound states are always discrete.
The energy eigenvalues and the corresponding wave functions, which are sometimes called the eigenfunctions, are obtained by solving the Schroedinger equation (1) under the boundary condition (5). Two examples are shown in the following figures, Fig. (B) and Fig. (C), the former of which is for the case of a square-well potential and the latter for a harmonic oscillator. For the details of the method to solve these problems, refer to an appropriate textbook of quantum mechanics.
The eigenstates in the
case of a square-well potential
The energy eigenvalues are represented by the heights of the horizontal levels which are obtained by solving the Schroedinger equation (1). The corresponding wave functions (thick solid curves) are also shown. The yellow-colored part denotes the potential wall. The number of the possible bound states depends on the value of .
The eigenstates of
a harmonic oscillator
The energy eigenvalues are represented by the heights of the horizontal levels which are obtained by solving the Schroedinger equation (1). The corresponding wave functions (thick solid curves) are also shown. The yellow-colored part denotes the potential wall.
The energy eigenvalues measured from the ground-state (0) are integral multiples of . This is a distinctive feature of the harmonic oscillator.
[The Reason of the Existence
of Discrete Eigenvalues]
We should pay attention to the reason why such discrete (step-like) eigenvalues appear. This is because a particle motion accompanies the wave function. In other words, the eigenstates with discrete energy eigenvalues occur because of the particle-wave duality. This is extremely surprising and never happens in the classical theory.
|Go back to the top page of Part 3. Go to the next page.|